High-Performance General Solver for Extremely Large-Scale Semidefinite Programming Problems

Semidefinite programming (SDP) is one of the most important problems among optimization problems at present. It is relevant to a wide range of fields such as combinatorial optimization, structural optimization, control theory, economics, quantum chemistry, sensor network location and data mining. The capability to solve extremely large-scale SDP problems will have a significant effect on the current and future applications of SDP. In 1995, Fujisawa et al. started the SDPA(Semidefinite programming algorithm) Project aimed at solving large-scale SDP problems with high numerical stability and accuracy. SDPA is one of the main codes to solve general SDPs. SDPARA is a parallel version of SDPA on multiple processors with distributed memory, and it replaces two major bottleneck parts (the generation of the Schur complement matrix and its Cholesky factorization) of SDPA by their parallel implementation. In particular, it has been successfully applied to combinatorial optimization and truss topology optimization. The new version of SDPARA (7.5.0-G) on a large-scale supercomputer called TSUBAME 2.0 at the Tokyo Institute of Technology has successfully been used to solve the largest SDP problem (which has over 1.48 million constraints), and created a new world record. Our implementation has also achieved 533 TFlops in double precision for large-scale Cholesky factorization using 2,720 CPUs and 4,080 GPUs.